INTERPRETING COVARIATE EFFECTS 37
2.7.3 Marginal vs. conditional effects
As indicated earlier in this chapter, the regression coefficients have distinctly
different interpretations in marginal and conditional models. The marginal
mean is defined as the mean response, averaged over the population from
which the sample is drawn. Regression coefficients in directly specified marginal
models capture the effects of covariates between groups of individuals defined
by differences in covariate values; these effects are sometimes referred to as
‘population-averaged’ effects (Zeger and Liang, 1992).
By contrast, regression coefficients in conditionally specified random ef-
fects models capture the effect of a covariate at the level of aggregation in the
random effect. For example, in model (2.5), random effects capture individual-
level variation; the regression coefficients β therefore represent covariate ef-
fects at the individual level. When the level of aggregation represented by
random effects is an individual or subject, the conditional regression param-
eters are sometimes referred toas‘subject-specific’ effects.
An important consideration in the interpretation of regression parameters
from models of repeated measures is whether the covariate in question varies
only between individuals, or whether it also varies within individuals (Fitz-
maurice et al., 2004). Two examples servetoillustrate the point. Consider
first the longitudinal studies of smoking cessation described in Section 1.4,
where the binary response is cessation status Y
ij
at measurement occasions
j =1,...,J.Furthersuppose that each individual is enrolled in one of two
treatment programs, denoted by X
i
∈{0, 1}.Arandom effects logistic re-
gression can be used to characterize a conditional or subject-specific effect
of X,
logit{E(Y
ij
| x
i
,b
i
,β)} = b
i
+ β
0
+ β
1
x
i
,
where (say) b
i
∼ N(0,σ
2
). The random effect b
i
hypothetically captures all
observable variation in the mean of Y
ij
that is not explained by x,andin
that sense two individuals with the same value of b
i
can be thought of as
perfectly matched with respect to unobserved individual-level covariate ef-
fects. Hence the coefficient β
1
can be interpreted as theeffect of treatment
on smoking cessation for two individuals with the same unobserved covariate
profile (i.e., thesamevalue of b
i
), or for two individuals whose predictors of
smoking cessation, other than treatment, are exactly the same. Here, as in
all random effects models, it is being assumed that unobserved characteristics
can be captured with a scalar value b
i
that follows a normal distribution. Both
are strong assumptions, and the ‘matching’ interpretation hinges critically on
correct specification of the model.
When X is a covariate that could potentially be manipulated (such as
treatment or environmental exposure), and if the study design is appropriate,
then β
1
sometimes can be interpreted as the causal effect of X on Y because
it is the conditional effect of X,givenacovariate measure that is technically